{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(integrals-1-building-blocks)=\n",
    "# Definite Integrals, Part 1: The Building Blocks\n",
    "\n",
    "*Updated 2023-11-07, correcting some typos.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**References:**\n",
    "\n",
    "- Sections 5.2.1 and 5.2.4 of Chapter 5 *Numerical Differentiation and Integration* in {cite}`Sauer`.\n",
    "- Sections 4.3 *Elements of Numerical Integration* of {cite}`Burden-Faires`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction\n",
    "\n",
    "The objective of this and several subsequent sections is to develop methods for approxmating a definite integral\n",
    "\n",
    "$$ I = \\int_a^b f(x) \\; dx $$\n",
    "\n",
    "This is arguably even more important than approximating derivatives, for several reasons;\n",
    "in particular, because there are many functions for which antiderivative formulas cannot be found,\n",
    "so that the result of the Fundamental Theorem of Calculus, that\n",
    "\n",
    "$$ \\int_a^b f(x) \\; dx = F(b) - F(a), \\text{ for } F \\text{ any antiderivative of } f$$\n",
    "\n",
    "does not help us.\n",
    "\n",
    "One core idea is to approximate the function $f$ by a polynomial (or several), and use its integral as an approximation.\n",
    "The two simplest possibilities here are approximating by a constant and by a straight line;\n",
    "here we explore the latter; th former wil be visited soon."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy import array, linspace, exp\n",
    "from matplotlib.pyplot import figure, plot, title, grid, legend"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Approximating with a single linear function: the Trapezoid Rule\n",
    "\n",
    "The idea is to approximate $f:[a, b] \\to \\Bbb{R}$ by collocation at the end points of this interval:\n",
    "\n",
    "$$ f(x) \\approx L(x) := \\frac{f(a)(b-x) + f(b)(x-a)}{b-a}, = f_{ave} (b-a) $$\n",
    "\n",
    "Then the approximation — which will be called $T_1$, for reasons that will becom clear soon — is\n",
    "\n",
    "$$I \\approx T_1 = \\int_a^b L(x) dx = \\frac{f(a) + f(b)}{2} (b-a) $$\n",
    "\n",
    "This can be interpreted as replacing $f(x)$ by $f_{ave}$ the average of the value at the end points,\n",
    "and inegrting that simple function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the example $f(x) = e^x$ on $[-1, 3]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 1\n",
    "b = 3\n",
    "def f(x): return exp(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 1008x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f_ave = (f(a) + f(b))/2\n",
    "x = linspace(a, b)\n",
    "figure(figsize=[14,10])\n",
    "plot(x, f(x))\n",
    "plot([a, a, b, b, a], [0, f(a), f(b), 0, 0], label=\"Trapezoid Rule\")\n",
    "plot([a, a, b, b, a], [0, f_ave, f_ave, 0, 0], '-.', label=\"Trapezoid Rule area\")\n",
    "legend()\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The approximation $T_1$ is the area of the orange trapezoid (hence the name!) which is also the area of the green rectangle."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Approximating with a constant: the Midpoint Rule\n",
    "\n",
    "The idea here is to approximate $f:[a, b] \\to \\Bbb{R}$ by its value at the midpoint of the interval,\n",
    "like the building blocks in a Riemann sum with the middel being the intuitive best choice of where to put the rectangle.\n",
    "\n",
    "$$ f(x) \\approx f_{mid} := f \\left(\\frac{a+b}{2}\\right) $$\n",
    "\n",
    "Then the approximation — which will be called $M_1$ — is\n",
    "\n",
    "$$ I \\approx M_1 = \\int_a^b f_{mid} \\, dx =  f \\left(\\frac{a+b}{2}\\right)(b-a) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the same example $f(x) = e^x$ on $[-1, 3]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1008x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f_mid = f((a+b)/2)\n",
    "figure(figsize=[14,10])\n",
    "plot(x, f(x))\n",
    "plot([a, a, b, b, a], [0, f_mid, f_mid, 0, 0], 'r', label=\"Midpoint Rule\")\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The approximation $M_1$ is the area of the red rectangle."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The two methods can be compared my combining these graphs:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1008x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f_mid = f((a+b)/2)\n",
    "figure(figsize=[14,10])\n",
    "plot(x, f(x))\n",
    "plot([a, a, b, b, a], [0, f(a), f(b), 0, 0], label=\"Trapezoid Rule\")\n",
    "plot([a, a, b, b, a], [0, f_ave, f_ave, 0, 0], '-.', label=\"Trapezoid Rule area\")\n",
    "plot([a, a, b, b, a], [0, f_mid, f_mid, 0, 0], 'r', label=\"Midpoint Rule\")\n",
    "legend()\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Error Formulas\n",
    "\n",
    "These graphs indicate that the trapezoid rule will over-estimate the error for this and any function that is convex up on the interval $[a, b]$.\n",
    "With closer examination it can perhaps be seen that the Midpoint Rule will instead underestimate in this situation, because\n",
    "its \"overshoot\" at left is less than its \"undershoot\" at right.\n",
    "\n",
    "We can derive error formulas that confirm this, and which are the basis for both practical error estimates and for deriving more accurate approximation methods.\n",
    "\n",
    "The first such method will be to use multiple small intervals instead of a single bigger one (using piecewise polynomial approximation) and for that, it is convenient to define $h = b-a$ which will become the parameter that we reduce in order to improve accuracy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:theorem} Error in the Trapezoid Rule, $T_1$\n",
    ":label: trapezoid-rule-error\n",
    "\n",
    "For a function $f$ that is twice differentiable on interval $[a, b]$, the error in the Trapezoid Rule is\n",
    "\n",
    "$$ \\int_a^b f(x) dx - T_1 = -\\frac{(b-a)^3}{12}f''(\\xi) \\quad \\text{for some} \\; \\xi \\in [a, b] $$\n",
    "\n",
    "It will be convenient to define $h := b-a$ so that this becomes\n",
    "\n",
    "$$ \\int_a^b f(x) dx - T_1 = -\\frac{h^3}{12}f''(\\xi) \\quad \\text{for some} \\; \\xi \\in [a, b]. $$\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:theorem} Error in the Midpoint Rule, $M_1$\n",
    ":label: midpoint-rule-error\n",
    "\n",
    "For a function $f$ that is twice differentiable on interval $[a, b]$ and again with $h=b-a$,\n",
    "the error in the Midpoint Rule is\n",
    "\n",
    "$$ \\int_a^b f(x) dx - M_1 = \\frac{h^3}{24}f''(\\xi) \\quad \\text{for some} \\; \\xi \\in [a, b] $$\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These will be verified below, using the error formulas for Taylor polynomials and collocation polynomials.\n",
    "\n",
    "For now, note that:\n",
    "- The results confirm that for a function that is convex up, the Trapezoid Rule overestimates and the Midpoint Rule underestimates.\n",
    "- The ratio of the errors is approximately $-2$. This will be used to get a better result by using a weighted average: *Simpson's Rule.*\n",
    "- The errors are $O(h^3)$. This opens the door to Richardson Extrapolation, as will be seen soon in the method of *Romberg Integration.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Proofs of these error results\n",
    "\n",
    "One side benefit of the following verifications is that they also offer illustrations of how the two fundamental error formulas help us: Taylor's Formula and its cousin the error formula for polynomial collocation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To help prove the above formulas, we introduce a result that also helps in various places later:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:theorem} The Integral Mean Value Theorem\n",
    ":label: integral-mean-value-theorem\n",
    "\n",
    "In an integral\n",
    "\n",
    "$$ \\int_a^b f(x) w(x) \\, dx  $$\n",
    "\n",
    "with $f$ continuous and the \"weight function\" $w(x)$ positive valued\n",
    "(actually, it is enough that $w(x) \\geq 0$ and it is not zero everyhere),\n",
    "there is a point $\\xi \\in [a,b]$ that gives a \"weighted average value\" for $f(x)$ in the sense that\n",
    "\n",
    "$$ \\int_a^b f(x) w(x) \\, dx = \\int_a^b f(\\xi) w(x) \\, dx, = f(\\xi) \\int_a^b w(x) \\, dx $$\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:proof} \n",
    "\n",
    "As $f$ is continuous on the closed, bounded interval $[a, b]$, the **Extreme Value Theorem** from calculus says that $f$ has a minimum $L$ and a maximum $H$ on this interval: $L \\leq f(x) \\leq H$.\n",
    "Since $w(x) \\geq 0$, this gives\n",
    "\n",
    "$$ L w(x) \\leq f(x) w(x) \\leq H w(x) $$\n",
    "\n",
    "and by integrating,\n",
    "\n",
    "$$ L \\int_a^b w(x) \\,dx \\leq \\int_a^b f(x) w(x) \\,dx \\leq H \\int_a^b w(x) \\,dx $$\n",
    "\n",
    "Dividing by $\\int_a^b w(x) \\,dx$ (which is positive),\n",
    "\n",
    "$$ L \\leq \\frac{\\int_a^b f(x) w(x) \\,dx}{\\int_a^b w(x) \\,dx} \\leq H $$\n",
    "\n",
    "and the Mean Value Theorem says that $f$ attains this value for some $\\xi \\in [L, H]$:\n",
    "\n",
    "$$ f(\\xi) = \\frac{\\int_a^b f(x) w(x) \\,dx}{\\int_a^b w(x) \\,dx} $$ (integral-mean-value-formula)\n",
    "\n",
    "Clearing the denominator gives the claimed result.\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:proof}\n",
    "(of {prf:ref}`trapezoid-rule-error`, {prf:ref}`the trapezoid rule error formula) <trapezoid-rule-error>`\n",
    "\n",
    "The function integrated to get the Trapezoid Rule is the linear collocating polynomial $L(x)$,\n",
    "and from the section {ref}`polynomial-collocation-error-formulas`, we have\n",
    "\n",
    "$$ f(x) - L(x) = \\frac{f''(\\xi_x)}{2}(x-a)(x-b) $$\n",
    "\n",
    "Integrating each side gives\n",
    "\n",
    "$$\n",
    "\\int_a^b (f(x) - L(x))\n",
    "= I - T_1\n",
    "= \\int_a^b \\frac{f''(\\xi_x)}{2}(x-a)(x-b) \\, dx\n",
    "$$\n",
    "\n",
    "To get around the complication that $\\xi_x$ depends on $x$ in an unknown way, use the Ingtgral Measn Vlaue Theorem with weight function\n",
    "$w(x) = (x-a)(b-x), \\geq 0$ for $a \\leq x \\leq b$.\n",
    "Then with $-f''$ as the function $f$ in {eq}`integral-mean-value-formula`:\n",
    "\n",
    "$$ I - T_1 = - \\int_a^b \\frac{f''(\\xi_x)}{2}(x-a)(b-x) \\, dx\n",
    "= - \\frac{f''(\\xi)}{2} \\int_a^b (x-a)(b-x) \\, dx $$\n",
    "\n",
    "A bit of calculus gives $\\displaystyle \\int_a^b (x-a)(b-x) \\, dx = \\frac{(b-a)^3}{6}$, so\n",
    "\n",
    "$$ I - T_1  = -\\frac{f''(\\xi)}{2} \\frac{(b-a)^3}{6} = -\\frac{f''(\\xi)}{12} (b-a)^3 = -\\frac{f''(\\xi)}{12} h^3, $$\n",
    "\n",
    "as advertised.\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:proof}\n",
    "(of {prf:ref}`midpoint-rule-error`, {prf:ref}`the midpoint rule error formula) <midpoint-rule-error>`\n",
    "\n",
    "For this, we can use Taylor's Theorem for the linear approximation\n",
    "\n",
    "$$ f(x) = f(c) + f'(c)(x-c) + \\frac{f''(\\xi_x)}{2}(x-c)^2 $$\n",
    "\n",
    "with $c = (a+b)/2$, the midpoint. That is,\n",
    "\n",
    "$$ f(x) - f(c) = f'(c) (x-c) + \\frac{f''(\\xi_x)}{2} (x-c)^2 $$\n",
    "\n",
    "and integrating each side gives\n",
    "\n",
    "$$\n",
    "\\int_a^b f(x) -  f(c) \\, dx\n",
    "= I - M_1\n",
    "= \\int_a^b \\left[ f'(c)(x-c) + \\frac{f''(\\xi_x)}{2}(x-c)^2 \\right] dx\n",
    "$$\n",
    "\n",
    "Here symmetry helps, by eliminating the first (potentialy biggest) term in the error:\n",
    "we use the fact that $a = c - h/2$ and $b = c + h/2$\n",
    "\n",
    "$$ \\int_a^b f'(c)(x-c) \\, dx = f'(c) \\int_{c-h/2}^{c + h/2} x-c \\, dx = \\left[ (x-c)^2/2 \\right]_{c-h/2}^{c + h/2} = (h/2)^2 - (h/2)^2 = 0 $$\n",
    "\n",
    "Thus the error simplifies to\n",
    "\n",
    "$$ I - M_1 = \\int_a^b \\frac{f''(\\xi_x)}{2}(x-c)^2 \\, dx $$\n",
    "\n",
    "and much as above, the Integral Mean Value Theorem can be used, this time with weight function $w(x) = (x-c)^2, \\geq 0$:\n",
    "\n",
    "$$ I - M_1 = \\frac{f''(\\xi)}{2} \\int_a^b (x-c)^2 \\, dx $$\n",
    "\n",
    "Another caluclus exercise:\n",
    "$\\displaystyle \\int_a^b (x-c)^2 dx = \\int_{-h/2}^{h/2} x^2 dx = \\left[x^3/3\\right]_{-h/2}^{h/2} = h^3/12$,\n",
    "so indeed,\n",
    "\n",
    "$$ I - M_1 = \\frac{f''(\\xi)}{24} h^3 $$\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(left-endpoint-rule)=\n",
    "## Appendix: Approximating a Definite Integral With the Left-hand Endpoint Rule\n",
    "\n",
    "An even simpler approximation of $\\int_a^b f(x)\\, dx$ is the *Left-hand Endpoint Rule*,\n",
    "probably seen in a calculus course.\n",
    "For a single interval, this uses the approximation\n",
    "\n",
    "$$ f(x) \\approx f(a) $$\n",
    "\n",
    "leading to\n",
    "\n",
    "$$I := \\int_a^b f(x)\\, dx \\approx L_1 :=  \\int_a^b f(a) \\, dx = f(a) (b-a) $$\n",
    "\n",
    "The correpsonding composite rule with $n$ sub-intervals of equal width $h = (b-a)/n$ is\n",
    "\n",
    "$$L_n = \\sum_{i=0}^{n-1} f(x_i) h, \\; = \\sum_{i=0}^{n-1} f(a + i h) h$$\n",
    "\n",
    "with $x_i = a + i h$ as before."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:theorem} Error in the Left-hand Endpoint Rule, $L_1$\n",
    ":label: error-left-endpoint-rule\n",
    "\n",
    "For a function $f$ that is differentiable on interval $[a, b]$, the error in the Left-hand Endpoint Rule is\n",
    "\n",
    "$$ \\int_a^b f(x) dx - L_1 = \\frac{(b-a)^2}{2}f'(\\xi), = \\frac{h^2}{2}f'(\\xi) \\quad \\text{for some} \\; \\xi \\in [a, b]n$$\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:proof}\n",
    "\n",
    "This time use Taylor's Theorem just for the constant approximation with center $a$:\n",
    "\n",
    "$$ f(x) = f(c) + f'(\\xi_x)(x-a) $$\n",
    "\n",
    "That is,\n",
    "\n",
    "$$ f(x) - f(a) = f'(\\xi_x)(x-a)$$\n",
    "\n",
    "so integrating each side gives\n",
    "\n",
    "$$\n",
    "\\int_a^b f(x) -  f(a) \\, dx\n",
    "= I - L_1\n",
    "= \\int_a^b  f'(\\xi_x)(x-a) dx\n",
    "$$\n",
    "\n",
    "Using the Integral Mean Value Theorem again, now with weight $w(x) = x-a$ gives\n",
    "\n",
    "$$ \\int_a^b f'(\\xi_x)(x-a) dx = f'(\\xi) \\int_a^b (x-a) dx = f'(\\xi) \\frac{(b-a)^2}{2} = \\frac{h^2}{2} f'(\\xi) \\; \\text{for some} \\; \\xi \\in [a, b] $$\n",
    "\n",
    "and inserting this into the previous formula gives the result.\n",
    "```"
   ]
  }
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